Name Class Date

6-1 Practice Form G

Solving Systems by Graphing

Solve each system by graphing. Check your solution.

1. y = x+ 3y = 4x 2

2. y = (1/2)x 2y = 3x + 5

3. y = (3/2)x + 6x + y = 1

4. y = 5xy = x 6

5. 3x + y = 5y = 7

6. y = 4x 6y = x + 9

7. y = (3/4)x 5 3x 4y = 20

8. y = (4/3)x 3y = (2/3)x + 3

9. y = (2/5)x 2y = x 5

10. Reasoning Can there be more than one point of intersection between the graphs of two linear equations? Why or why not?

11. Reasoning If the graphs of the equations in a system of linear equations coincide with each other, what does that tell you about the solution of the system? Explain.

12. Writing Explain the method used to graph a line using the slope and y-intercept.

13. Reasoning If the ordered pair (3, 2) satisfies one of the two linear equations in a system, how can you tell whether the point satisfies the other equation of the system? Explain.

14. Writing If the graphs of two lines in a system do not intersect at any point, what can you conclude about the solution of the system? Why? Explain.

15. Reasoning Without graphing, decide whether the following system of linear equations has one solution, infinitely many solutions or no solution. Explain.

y = 3x 56x = 2y + 10

16. Five years from now, a father’s age will be three times his son’s age, and 5 years ago, he was seven times as old as his son was. What are their present ages?

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Name Class Date

6-1 Practice (continued) Form G

Solving Systems by Graphing

17. The denominator of a fraction is greater than its numerator by 9. If 7 is subtracted from both its numerator and denominator, the new fraction equals

.

What is the original frac

tion?

18. The sum of the distances two hikers walked is 53 mi, and the difference is 25 mi. What are the distances?

19. The result of dividing a two-digit number by the number with its digits reversed

is

If

the sum of the digits is 12, what is the number?

Solve each system by graphing. Tell whether the system has one solution, infinitely many solutions, or no solution.

20. y = 3x + 5x + y = 3

21. y = 2x + 1y = 4x + 7

23. y = 2x + 1y = (-2/3)x + 5

24. y = 3x + 23x + y =

26. y = (1/2)x 6y = (1/4)x

27. y = 6x +2 + y =

29. 18x 3y = 21y = 6x + 7

30. y = 5x 6x + y =

32. The measure of one of the angles of a triangle is 35. The sum of the measures of the other two angles is 145 and the difference between their measures is 15. What are the measures of the unknown angles?

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6-2 Practice Form G

Solving Systems Using Substitution

Solve each system by substitution. Check your solution.

1. x = y 2. y = x + 4 3. y = 2x 10x + 2y = 3 y = 3x 2y = x 8

4. 2y = x + 1 5. x + 2y = 14 6. 2x 3y = 132x y = 7 y = 3x 14

7. 3x 2y = 5.5 8. 6x 4y = 54 9.

x + 3y = 7.5 9x + 2y = 69 2x y = 5

10. Writing How do you know that substitution gives the answer to a system of equations? Explain.

11. Reasoning With the substitution method, which variable should you solve for first? Explain.

12. Writing How can you use substitution method to solve a system of equations that does not have a variable with a coefficient of 1 or –1?

13. Writing When solving the system of equations using

substitution, which variable will you solve for and which equation will you use to substitute into?

14. Reasoning Can you tell that there is no solution for a system by just looking at the equations? Explain and give an example.

15. If the difference in the side lengths of two squares is 10, and the sum of the side lengths is 18, what are the side lengths?

16. A shopper purchased 8 T-Shirts and 5 pairs pants for $220. The next day, he purchased 5 T-shirts and 1 pair of pants for $112. How much does each T-shirt and each pair of pants cost?

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Name Class Date

6-2 Practice (continued) Form G

Solving Systems Using Substitution

17.A student bought 1 box of crayons and 5 reams of pape

r for $54. She bought 5 boxes of crayons and 3 reams o

f paper for $50. What is the cost of each box of crayo

ns and each ream of paper?

18.

Suppose you got 8 mangoes and 3 apples for $18 and 3 mangoes and 5 apples

for $14.50. How much does each mango and each apple cost?

19.A shopper purchased 4 tables and

2 chairs for $200 and 2 tables and 7 chairs

for $400.

What is the cost of each table and each chair?

20.If the lengt

h of the rectangle is twice the width, and the perimete

r of the rectangle is 30 cm, what is length and width o

f the rectangle?

21.The population of a city is 2,500. If th

e number of males is 240 more than the number of female

s, how many males and females are there in the city?

S

olve each system by substitution. Tell whether the syst

em has one solution, infinitely many solutions, or no s

olution.

22. 7x + 2y = 13 23. x 9y =

3x 8y = 23 6x + y =

25. x 2y 1 = 0 26. y = 8x y 5x + 14 = 0 x + 3y =

28. 5x 9y = 29 29. 2x = 3y 12x + y = 47 3x + y =

31. x = 6y + 16 32. 4x y 4 = 09x 2y = 12 3x + 2y

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6-3 Practice Form G

Solving Systems Using Elimination

Solve each system using elimination.

1. x+ y = 2 2. x + 2y = 3 3. 2x y = 4

x y = 4 x y = 6 3x y = 2

4. x 2y = 2 5. x 3y = 3 6. x + 2y = 4

x + y = 3 2x + 3y = 5 x + y = 2

7. 3x 2y = 8 8. x 2y = 3 9. 2x 4y = 6

2x 2y = 5 3x y = 2 x y = 1

10. Writing For the system , which variable should you eliminate

first and why? How will you eliminate that variable?

11. Open-Ended If you do not have equal coefficients for both variables, can you still use the elimination method? Explain.

12. In a class, 45 students take the SAT exam. The number of boys is 8 more than the number of girls.

a. Write a system that models the above situation.b. Do you need to multiply any of the equations by a constant? If so, which

equation and what is the constant?

13. Open-Ended Write a system for which using the elimination method to solve the system is easier than the substitution method. Explain.

14. Error Analysis A student solved a system of linear equations using the elimination method as follows. Describe and correct the error made by the student.

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Multiply equation 1 by 2.

Multiply equation 2 by 3.

Add the equations.

Divide by 7.

Name Class Date

6-3 Practice (continued) Form G

Solving Systems Using Elimination

15. A farm raises a total of 220 chickens and pigs. The n

umber of legs of the stock in the farm totals 520. How

many chickens and pigs are at the farm?

16. You drive a ca

r that runs on ethanol and gas. You have a 20-gallon ta

nk to fill and you can buy fuel that is either 25 perce

nt ethanol or 85 percent ethanol. How much of each type

of fuel should you buy to fill your tank so that it is

50 percent ethanol?

17. Your math test has 38 questions a

nd is worth 200 points. The test consists of multiple-c

hoice questions worth 4 points each and open-ended ques

tions worth 20 points each. How many of each type of qu

estion are there?

18. A student bought 3 boxes of pencils

and 2 boxes of pens for $6. He then bought 2 boxes of p

encils and 4 boxes of pens for $8. Find the cost of eac

h box of pencils and each box of pens.

Solve each sys

tem using elimination. Tell whether the system has one

solution, infinitely many solutions, or no solution.

19. x 3y = 27

2x = 6y 14 x + 3y = 4

22. 5x + y = 15 23. 3x = 4y 53y = 15x + 6 12y = 9x +

25. x + 2y = 4 26. x + y = 23x + 2y = 4 x – y = 4

28. 4x 3y = 5 29. x 3y = 13x 2y = 8 2x + 2y =

31.How is the multiplication or division propert

y of equality used in the elimination method? Are the p

roperties always needed? Explain.

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6-4 Practice Form G

Applications of Linear Systems

Solve each word problem

1. You have $6000 to invest in two stock funds. The first fund pays 5% annual interest and the second account pays 9% annual interest. If after a year you have made $380 in interest, how much money did you invest in each account?

2. During a sale at the local department store, you buy three sweatshirts and two pairs of sweatpants for $85.50. Later you return to the same store and buy three more sweatshirts and four more pairs of sweatpants for $123. What is the sale price of each sweatshirt and each pair of sweatpants?

3. The sum of two numbers is 27. The larger number is 3 more than the smaller number. What are the two numbers?

4. One plane at 520 feet is ascending at a rate of 40 feet per minute, while another plane at 3800 feet is descending at a rate of 120 feet per minute. How long will it take the two planes to be at the same altitude?

5. The perimeter of a rectangle is 24 in. and its length is 3 times its width. What are the length and the width of the rectangle?

6. You are getting ready to move and have asked some friends to help. For lunch, you buy the following sandwiches at the local deli for $30: six tuna sandwiches and six turkey sandwiches. Later at night, everyone is hungry again and you buy four tuna sandwiches and eight turkey sandwiches for $30.60. What is the price of each sandwich?

7. You have a cable plan that costs $39 a month for a basic plan plus one movie channel. Your friend has the same basic plan plan plus two movie channels for $45.50. What is the basic plan charge that you both pay?

8. At an all-you-can-eat barbeque fundraiser that you are sponsoring, adults pay $6 for a dinner and children pay $4 for a dinner. 212 people attend and you raise $1128. What is the total number of adults and the total number of children attending?

a. What is a system of equations that you can use to solve this problem?

b. What method would you use to solve the system? Why?

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Name Class Date

6-4 Practice (continued) Form G

Applications of Linear Systems

Solve each system. Explain why you chose the method you used.

9. 2y = x + 1 10. 6x – 4y = 54

2x y = 7 9x + 2y = 69

11. 3x 2y = 8 12. 2x y = 42x 2y = 5 3x y = 2

13. 2x 3y = 13 14. x 3y = 3

2x + 3y = 5

15. Open-Ended What are three differences between an inconsistent system and a consistent and independent system? Explain.

16. Reasoning One number is 4 less than 3 times a second number. If 3 more than two times the first number is decreased by two times the second, the result is 11. What are both numbers?

17. Error Analysis In Exercise 16, what kind of errors are likely to occur when solving the problem?

18. A plane leaves Chicago and flies 750 miles to New York. If it takes 2.5 hours to get to New York flying against the wind, but only 2 hours to fly back to Chicago, what is the plane’s rate of speed and what is the wind speed?

19. A coin bank has 250 coins, dimes and quarters, worth $39.25. How many of each type of coin are there?

20. In 4 years, a mother will be 5 times as old as her daughter. At present, the mother is 9 times as old as the daughter. How old are the mother and the daughter today?

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